We all love the Super Bowl, not only for the football (go Patriots!), but also for those hilarious beer commercials, in which supermodels abound and dogs can be trained to bring you a “cold one”. But the companies who pay for these Super Bowl advertisements are not here to entertain us. So why, then, are they willing to spend millions of dollars to develop and broadcast their ads? I have heard many fellow TV viewers say, “That was funny, but it doesn’t make me want to buy (insert product name here).” So, why do these companies spend so much on advertising, when it doesn’t seem to affect our purchasing behavior? The answer can be found using game theory.

To answer this question, we will analyze the light beer market: an industry whose advertisements are inescapable during football games. Our two players in this “advertising game” will be Bud Light and Coors Light. (Anheuser-Busch, Bud’s parent company, spent $24 million on Super Bowl ads in 2011!) For the purposes of this analysis, we will say that each brewer is trying to decide whether to spend $50 million or $10 million on advertising during the playoff football games leading up to the Super Bowl. We will also assume that there is $500 million of total revenue in the light beer market, and the total profit will equal total revenue minus the advertising cost. If both Bud Light and Coors Light decide to spend only $10 million, they will each make a profit of $240 million. If one company decides to spend $50 million while the other spends only $10 million, then the one that spends more will profit $340 million while the other makes only $100 million. Finally, if both companies decide to spend $50 million on ads, they will each profit $200 million. As we can see, the profit of each company is dependent on the actions of the other, and when they both decide to spend $50 million, their extra advertisements effectively cancel one another out.

Here is a table of the payoffs of each company. In each cell, the first number is Bud Light’s profits and the second number is Coors Light’s.

The next question to ask is, where is the Nash equilibrium? We can see that each company has a dominant strategy. If I am in charge of advertising at Coors, I can see that I will get a greater profit if I spend $50 million on ads, whether Bud decides to spend $10 million or $50 million. Similarly, Bud will maximize profit by spending $50 million regardless of what Coors does. Therefore, the advertising game reaches a Nash equilibrium with each brewer spending $5o million on ads and each netting a $200 million profit. “But wait,” you might say, “this is not the optimal profit for either company!” As we see from the payoff table, both Bud Light and Coors Light would benefit if they made a gentleman’s agreement to spend only $10 million, and receive a higher payoff of $240 million due to the savings in advertising. However, this is not a Nash equilibrium. At this state, either company could greatly increase profits by defecting from the agreement and spending $50 million. This is an example of the Braess paradox, which means the equilibrium is not the best outcome for both parties. This paradox is the reason why we are barraged with ads during the time-outs of football games.